This paper develops a mixed finite element – smoothing particle method for violent water-structure interactions involving freak waves and separation between structure and water. The structure undergoes a large rigid motion of 6 DOF with a small elastic deformation, so that its elastic displacement relative to the rigid motion can be represented in a mode summation based on FE analysis. The water is assumed inviscid - incompressible and its motion governed by nonlinear N-S equation. On the coupling interface where no FS separation happens, the equilibrium and consistence conditions are required. The numerical iteration process is suggested to solve the nonlinear FSI equations, and validation examples are shown a good agreement with available experiment results

Djidjeli, Kamal

Sun, zhe

Xing, Jing

Southampton (e-Prints Soton)

  a) Corresponding author. Email: jtxing@soton.ac.uk.  XXIV ICTAM, 21-26 August 2016, Montreal, Canada  MIXED FE-SP METHOD FOR NONLINEAR STRUCTURE-WATER INTERACTIONS WITH FREAK WAVES Jing Tang Xing1a), Kamal Djidjeli2 & Zhe SUN 1,2 1FSI Group, FEE, University of Southampton, Southampton, UK 2CED Group, FEE, University of Southampton, Southampton, UK   Summary This paper develops a mixed finite element – smoothing particle method for violent water-structure interactions involving freak waves and separation between structure and water. The structure undergoes a large rigid motion of 6 DOF with a small elastic deformation, so that its elastic displacement relative to the rigid motion can be represented in a mode summation based on FE analysis. The water is assumed inviscid - incompressible and its motion governed by nonlinear N-S equation. On the coupling interface where no FS separation happens, the equilibrium and consistence conditions are required. The numerical iteration process is suggested to solve the nonlinear FSI equations, and validation examples are shown a good agreement with available experiment results. GOVERNING EQUATIONS As shown in Fig. 1, a sold in a material domains of outside boundary normal vector i , on which a traction force iTˆ and a displacement iUˆ are respectively given on boundary TS and US , interacts on its wet interface   with water domain f , of which the dynamic pressure 0p on free surface f , 0iv on b and possible given velocity ivˆ  on boundaries v or .v Three coordinate systems are defined: 321xxxo with positive 3x in the vertical direction is fixed in the space, 321 yyyO , of which axis iyO  parallels axis ixo, (i = 1,2,3), is a moving system fixed at the mass centre O of solid, and 321XXXO  is a material system. A material pointjX at time 0t moves to a new position jx at time t , and its motion is denoted by a summation of a translation iu of mass centre, rigid rotation iabout three axes and small elastic displacement jU of structure. Using the notations of Cartesian tensors and summation convention [1], such as Kronecker Delta ij and permutation symbolijke , and denoting the rigid rotation by a tensor )(kij with its time derivative kjmimkije    and partial derivativekjimkmije   / , we can derive the dynamic equations of the system as follows.    Solid: The position and velocity of particlejX at time t respectively are             .)(,jijjjkjmimkijijjijjijiijijjijiiUUXeuUUXuxUXux                (1) Introducing a transformationijH with its related generalized coordinateiU~ , which may be a finite element interpolation function matrixijH[2] and node displacement vector, or a mode function matrix and generalised mode coordinate vector [3], we denote the elastic displacement UHU ii~ and define ,~  s jsj dHH     s njsjn dHHV ,  ,  s njsjn dHXS   ,~ jjVM  ),,,2,1,( N so that the kinetic and potential energy of body are respectively calculated in the forms .,~,,ˆˆ,ˆ~,ˆˆ,~~)(~)](2[)(),(~,~),(,~,~~ˆ~~ˆ~~~~~5.0,~)(,~~),(},~2~22~~~~5.0,,33dSpXdSpHPdSpPdSTXdSTHFdSTFUUVVUSSSJJUJUVeUCdHEHKPUPuFUFuUHβgugMUUKHBUHeUAUCUuBuAUUMJuuMTllpllllTS llTS llTS lljnsnrjjjrsnjljnsnrjjjrsjnsnrjrsnnrsjninkjrkirlks ijkljiplrlrlrlrjijirrnrnrjkjrskrsrrrrsrrssrrssrrs (2) Here M  is total mass of solid, M~is generalized mass matrix,lFˆ , ˆ andrF~ are resultant force, moment and generalised force, while the FSI forces involving fluid pressure p denotes by the last column in Eq. 3. We derive the dynamic equations of solid motion by using Hamilton equation [3], ,0///)/( iiiqqTdtqTd  therefore, we have PPuPFFuFHβgUHβggMUuUCUJUACBuJAUuKUuCBCJABAUuMCBCJABAMlrmlmlrllrmlmlrljijijrijirssllllsllrllrlslrlslrsllssssssrrssrrrsssssrrssrrrsrs~/~ˆ/ˆ~~~/~~/~/~5.0~////~5.0/000~~00000000~0~0~~~333. (3)  Fluid: The water motion is governed by Lagrangian Navier-Stokes equation for incompressible and inviscid fluid in association with corresponding boundary conditions [4-6] .on  ,ˆ;on  ,0;on  ,,)(;in  ,0,/vf,3,,3iiiiiiiiiiiffiifiiivvpUvpUgvpgv     (4) PARTITIONED ITERATION SOLUTION    Equations (3) and (4) include 3 rigid translationssu , 3 rigid rotationss and N generalised coordinatesU~ , in which the rigid motions could be large but the elastic deformation is assumed small, so that the second order quantity ofU~ are neglected. It is convenience to use a commercial FE code to obtain the natural modes and frequencies of the solid as the input data for the code designed for the method proposed in this paper. The partitioned iteration is suggested to solve these FSI equations [7-8], starting from initial time t = 0, at which the state of the system are given, to calculate the solution at time tt  in the following steps. i) Initial calculations: FE mode dada as input to generate related matrices; initial position, velocity, acceleration and pressure of the system. ii) Fluid SPM solver: based on the state of solid at time t to deliver fluid variables at time tt   using the projection [9] method: a) with no pressure effect to calculate an intermediate velocityiv~  and location ix~ of particles, ;~~,~3 itiiitiivtxxtgvv   b) a pressure Poisson equation is solved to obtain the pressure,),/()(/~ 200,,tnnntvp tfiifttii   |,||/| ,00 tiit vtnnn   where 0 1, respectively for   ,0)-( ,0 tiit vnn and,0)-(,0 tiit vnn  to identify compressed or expanded state of fluid; c) fluid velocity and position at time tt  are calculatedttitittifttiittitvxxtpvv   ,/~, . iii) Solid solver: based on the fluid pressure to solve solid equation by time integration iteration to obtain the state variables of the solid. iv) Convergence check: calculating the displacement error norm on FSI interface to see ifdSUU titti|| then go to next time step tt  2 , otherwise, to repeat process ii) and iii) until convergence reached. A calculation flow chat is shown in Fig. 2, where n denotes an FSI iteration number and other notations are the state variables used in the computer code, for which some SPM particular treatment techniques, such as neighbour particle searching, particle interactions with shifting and collision handling, free surface particle identification and FS surface separation flag are considered [10-13].         x3y3X3x1y1X1x2y2X2y3y1y2X1X2X3OOpPxx uΓfΓv2Γv1ΓbvρfgρggΩsΣ θ2θ3θ1ΩfFluid field modelled by particle methodt=0t=t1^T      Fluid Solver(Modified MPS)Structure Solver(CRMS)Yest=tn t=tn-1t=tn+1No𝑫𝑛   𝜞 Σ,𝑖+1𝑛  Convergence test𝑫𝑛−1 𝑫 𝑖+1𝑛  𝜞𝛴,𝑖+1𝑛  𝑫𝑖+1𝑛  𝜞 𝛴,0𝑛+1 𝜞𝛴,0𝑛+1 𝜞 𝛴,0𝑛+1 𝑫 𝑛  𝑫 𝑛  𝜞 𝛴,0𝑛  𝜞𝛴,0𝑛  𝜞 𝛴,0𝑛  𝑫 𝑛−1 𝑫 𝑛−1 𝒑𝛤,𝑖𝑛                              Fig.1 Scheme of FSI system                  Fig. 2 Flow chart of FSI partitioned iteration process VALIDATION AND APPLICATIONS     To validate the developed method, the numerical simulations [10-12] of rigid and elastic wedge dropping on the water surface have been completed and compared with available experiments [14-16], which are shown in Fig. 3.       0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.300.10.20.30.40.50.60.70.8x (m)y (m)294t=0.04s     (i)                             (ii)                            (iii)                           (iv)                        (v)  Fig. 3 Wedge dropping simulations: (i) rigid one comparison with experiment photo [14][10]; (ii) elastic test arrangement [16]; (iii) numerical picture at time 0.04s [11-12]; (iv) comparison of strains at two points from wedge tip with experiments a) 30mm, b) 120mm [15]; (v) comparison of accelerations of elastic wedge with experiments a) case 1 (4.29m/s), b) case 2 (5.57m/s) [16].    It has been shown a very good agreement with experiments, which demonstrates that the proposed method is attractive to deal with strong nonlinear FSI involving freak waves or FS separations, which is difficult to be solved by mesh based methods. This type of problems have an important background in marine engineering, such as green water impacts concerned by ship designs and aircrafts landing on the water. The developed method and its integrated computer code is still developing to tackle more engineering application cases discussed in many books, such as [17-19]. It is noticed that the system involves chaos and bifurcations that may be met in FSI numerical simulations, which requires further investigations using nonlinear dynamic approaches, such as the developed energy flow method [20].      References [1] Fung Y.C.: A first course in continuum mechanics. Prentice-Hall, 1977. [2] Bathe K. J.: Finite Element Procedures in Engineering Analysis. Prentice-Hall, 1996. [3] Meirovitch L.: Principles and techniques of vibrations. Prentice-Hall, 1997. [4] Monaghan J J. 1992. Smoothed particle hydrodynamics. Annual Review of Astr. and Astrophysics, 30:543-574. [5] Monaghan, :Simulating Free Surface Flows with SPH. Journal of Computational Physics 110: 399-406, 1994. [6] Liu G. R., Liu M. B.: Smoothed particle hydrodynamics. World Scientific Publishing Co. 2003. [7] Xing J. T., Price W. G., Chen Y. G.: A mixed finite element - finite difference method for nonlinear fluid-structure interaction dynamics, Part I: rigid structure-fluid interaction. Proc. Royal Soc A 459: 2399-2430. 2003 [8] Xing J.T.: Developments of numerical methods for linear and nonlinear fluid-solid interaction dynamics with applications. Advances in Mechanics (in print), 2016. [9] Chorin A.J.: A numerical method for solving incompressible viscous flow problems. Journal of Computational Physics 2: 12-26, 1967. [10] Sun F.: Investigations of smoothed particle hydrodynamics method for nonlinear fluid-rigid body interaction dynamics. PhD thesis, FEE, University of Southampton, 2013. [11] Sun Z., Djidjeli K., Xing J.T., Cheng F.: Modified MPS method for the 2D Fluid Structure Interaction problems with free surface. Computers & Fluids 122: 47-65, 2015. [12] Sun Z., Djidjeli K., Xing J.T., Cheng F.: Coupled MPS-modal superposition method for 2D nonlinear fluid-structure interaction problems with free surface. Journal of Fluids & Structures 61:295-323, 2016. [13] Javed A., Djidjeli K., Xing J. T.: Shape adaptive RBF-FD implicit scheme for incompressible viscous Navier–Stokes equations. Computer & Fluids, 89: 38-52, 2014. [14] Zhao R., Faltinsen O., Aarsnes J.: Water entry of arbitrary two-dimensional sections with and without flow separation. 21st Symposium on Naval Hydrodynamics,1997. [15] Panciroli R., Abrate S., Minak G., Zucchelli A.: Hydroelasticity in water-entry problems: comparison between experimental and SPH results. Comp Struc, 94: 532-539, 2012 [16] Panciroli R.: Hydroelastic Impacts of Deformable Wedges. Solid Mechanics and Its Applications 192: 1-45, 2013. [17] Morand H. J. P., Ohayon R.: Fluid structure interaction. John Wiley and Sons, Chichester, 1995. [18] Païdoussis M. P.: Fluid-Structure Interactions: Slender Structures and Axial Flow. Academic Press. 2013. [19] Païdoussis M. P., Price S. J., Langre E. D.: Fluid-Structure Interactions: Cross-Flow-Induced Instabilities. Cambridge University. 2011. [20] Xing J.T.: Energy flow theory of nonlinear dynamical systems with applications. Springer, 2015. 

Mixed FE-SP method for nonlinear structure-water interactions with freak waves

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